# Search Results

## You are looking at 1 - 10 of 10 items for :

• "11P32"
• Refine by Access: All Content
Clear All

## Exceptional set of a representation with fractional powers

Acta Mathematica Hungarica
Authors:
E. Balanzario
,
M. Garaev
, and
R. Zuazua

## Abstract

We prove that for any given c, 1 < c < 17/11, almost all natural numbers are representable in the form [x c] + [p c], where x is a natural number and p is a prime.

Restricted access

## On the slim exceptional set for the Lagrange four squares theorem

Acta Mathematica Hungarica
Authors:
Yingchun Cai
and
Minggao Lu

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E 3(N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P 5. Then we have E 3(N)≪log1053 N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P 11 in place of P 5.

Restricted access

## Hua’s theorem with nine almost equal prime variables

Acta Mathematica Hungarica
Authors:
G. Lü
and
Y. Xu

## Abstract

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N = p_1^3 + \cdots + p_9^3 with \left| {p_j - \sqrt[3]{{N/9}}} \right| \leqq U = N^{\tfrac{1} {3} - \tfrac{1} {{198}} + \varepsilon }$$ \end{document}
, where p j are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

Restricted access

## Sums of almost equal prime cubes

Acta Mathematica Hungarica
Author:
F. Zhao

## Abstract

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p 1 3 + ... + p j 3 with |p i − (N/j)1/3| ≦
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N^{1/3 - \delta _j + \varepsilon }$$ \end{document}
(1 ≦ ij), for δ j = 1/45; 1/30; 1/25; 2/45, respectively.
Restricted access

## An improvement on Waring–Goldbach problem for unlike powers

Acta Mathematica Hungarica
Author:
Zhixin Liu

## Abstract

Let p i be prime numbers. In this paper, it is proved that for any integer k≧5, with at most exceptions, all positive even integers up to N can be expressed in the form . This improves the result for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form . This method can also be used for some other similar forms.

Restricted access

## On a generalization of Hua’s theorem with five squares of primes

Acta Mathematica Hungarica
Authors:
G. Lü
and
H. Sun

## Abstract

We sharpen Hua’s theorem with five squares of primes by proving that every sufficiently large integer N congruent to 5 modulo 24 can be written in the form
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + p_5^2$$ \end{document}
with p 1
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N^{\tfrac{{49}} {{288}}}$$ \end{document}
.
Restricted access

## Difference sets and shifted primes

Acta Mathematica Hungarica
Author:
J. Lucier

## Abstract

We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)c(log log log log log n)) for some absolute c > 0.

Restricted access

## On the ternary goldbach problem with primes in independent arithmetic progressions

Acta Mathematica Hungarica
Author:
K. Halupczok

## Abstract

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q 1 = Q 2:= n 1/2(log n)ϑ and Q 3:= (log n) θ . Then for all q 3Q 3, all reduced residues a 3 mod q 3, almost all q 2Q 2, all admissible residues a 2 mod q 2, almost all q 1Q 1 and all admissible residues a 1 mod q 1, there exists a representation n = p 1 + p 2 + p 3 with primes p i a i (q i ), i = 1, 2, 3.

Restricted access

## On the sum of a prime and a k-th power of prime in short intervals

Acta Mathematica Hungarica
Author:
Y. C. Wang

## Abstract

Let denote the set {n∣2|n, ∀ p>2 with p−1|k}. We prove that when , almost all integers can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when , almost all integers n∊(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.

Restricted access

## On unlike powers of primes and powers of 2

Acta Mathematica Hungarica
Authors:
Zhixin Liu
and
Guangshi Lü

## Abstract

It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.

Restricted access